So, this is a specific task that states:
We're given a rectangular conductive contour with known dimensions $a=5cm$ and $b=7.5cm$ like in the picture. At the moment, the distance between the contour and a very long wire conductor, with known current $I=2A$, is $c=2.5cm$. The resistance of rectangular contour is $R=2\Omega$. At one point, the contour starts moving to the right (staying in the plane of the picture at the same time) and stops when it moves the distance $b$. What's the total charge that travels through the contour from the beginning moment to when it stops moving?
So, what I did first is stated that the magnetic field around the long wire conductor at some distance $r$ is:
$$\oint_C\vec{B}d\vec{l}=\mu_0\sum_C I$$
$$\oint_C B\cdot dl\cdot \cos<(\vec{B},d\vec{l})=\mu_0\sum_C I$$
$$\oint_C Bdl=\mu_0I$$
$$B\cdot 2\pi r=\mu_0 I$$
$$B=\frac{\mu_0 I}{2\pi r}$$
Now the magnetic flux through the surface of the rectangle would be:
$$\Phi=\int_S \vec{B}d\vec{S}=\int_S B\cdot dS\cdot cos<(\vec{B},d\vec{S})=\int_S B\cdot dS=$$
$$=\int_{c}^{c+b}\frac{\mu_0 I}{2\pi r}\cdot a\cdot dr=\frac{\mu_0 I a}{2\pi}\ln{\frac{c+b}{c}}$$
Now, what confuses me is that I don't know the exact time interval the contour traveled from the first to second position, nor do I understand if it's static or dynamic induction, or both that occur during the movement. I had the idea of calculating the electromagnetic induction that occurred during the movement, then calculating the current due to induction as: $$I_{ind}=\frac{e_{ind}}{R}$$ and then calculating the total charge that went through the contour as: $$Q=\int_{\Delta t}Idt$$
But for all of that I need the time interval in which the contour moved, or even the speed at which it moved, from which I could calculate the time.
Can this be solved in any other way?
